Modelling Claim Development Processes - Actuaries Institute Wang_Claim... · 2010-05-18 ·...

Modelling Claim Development Processes

Prepared by Bo Wang

Presented to the Institute of Actuaries of Australia

16th General Insurance Seminar 9-12 November 2008

Coolum, Australia

This paper has been prepared for the Institute of Actuaries of Australia’s (Institute) 16th General Insurance Seminar 2008.

The Institute Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is

not responsible for those opinions.

Bo Wang

The Institute will ensure that all reproductions of the paper acknowledge the author/s as the

author/s, and include the above copyright statement:

The Institute of Actuaries of Australia

Level 7 Challis House 4 Martin Place

Sydney NSW Australia 2000

Telephone: +61 2 9233 3466 Facsimile: +61 2 9233 3446

Email: [email protected] Website: www.actuaries.asn.au


1

Abstract

Pricing for long-tailed products has always being problematic. Take the example of CTP,

where claims can take more than ten years to settle, especially for injured children. It is

not unusual to have active claims outnumber finalised claims; and, typically, the

inclusion of active claims can bias the results, and equally so for the exclusion of active

claims.

This paper proposes a framework that can be used to analyse active claims side by side

with finalised claims for pricing purposes. This is done by modelling how a claim evolves

over time so the ultimate claim size can be inferred from the various attributes of a claim.

A positive side effect of this approach is that outstanding claims liability can be estimated

in the model framework using a simulation approach. However, reserving is not the

motivation of this paper.

The claim development process is broken down into the various component processes – a

claim status process that takes on binary values; a delay process that models the time

between each successive update to the incurred claims cost variable; a binary process that

models whether the claims cost is revised upwards or downwards at each successive

update; a positive process that models the size of the revision. These processes are

modelled with appropriate distributions and explanatory variables such as injury

severity, legal representation, etc.

Effectively, time is measured by the number of changes made to the estimate of the

claims cost from the reporting of the claim, making this approach analogous to the

operational time approach used in other reserving techniques. This approach can also be

used to account for case estimation policy changes, when a certain effect can be built into

the claim cost changes occurring within a specific timeframe.

When combined with an appropriate pricing framework, this approach will provide

more robust inferences based on all available claims information, rather than only a

subset.


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1 Introduction

1.1 Background

One of the fundamental issues that need to be dealt with in a pricing exercise is to project

claims to their ultimate values. The estimation of the claim costs for the underwriting

period under consideration would involve analysis of the claim cost patterns in the recent

past. However, the existence of open claims makes this difficult.

Open claims represent an incomplete view of the claims experience and usually cannot

be analysed in conjunction with the finalised claims. Typically, case estimates on the

open claims do not represent the mean of their ultimate outcomes and are typically

biased. Measures of average claim size or average claims cost per policy generally show

significant trends for the most recent accident periods due to the distortions caused by

these open claims. This issue is most evident in long-tailed products, where a significant

proportion of the reported claims remains open.

There are usually a few solutions to this issue. Firstly, the accident periods where open

claims present too significant an issue can simply be excluded. Then, the historical

accident periods where open claims are no longer an issue can be inflated appropriately

to the underwriting period. However, this method fails when the product is undergoing

change and the most recent experience is crucial in the analysis of claims experience.

Secondly, the pricing exercise can leverage off the actuarial outstanding claims valuation.

The valuation would typically contain outstanding-to-case estimate ratios for each of the

accident periods. The claims are then “grossed up” according to their stage of

development. However, this approach is across the board and fails when various

segments of the claims develop differently. For example, claims from different

occupations for a professional liability portfolio may have a different development

pattern.

This paper proposes a novel way of modelling claims, using characteristic information of

individual claims to model the complete claim cost evolution process. Claim cost

evolution refers to the history of the estimate incurred cost (payments to date plus case

estimate) from when the claim is reported to when it is finalised, change by change.

By modelling claims individually, this approach falls into the category of “Individual

claims models” (Taylor et al., 2006). The main difference between this approach and

others in the literature is that the claim evolution is modelled rather than the final result.

Too often claim modelling techniques concentrate on the end result, that is, the ultimate

claim size, neglecting a vast amount of information contained in the development of the

claims. This paper takes the approach of modelling claim development processes by

studying how the incurred claims cost changes whenever the case estimate is updated.

One major issue that researchers noted with this category of models is the usage of

“dynamic stochastic variables” (Taylor et al., 2006; Prevett and Gifford, 2007). While

Taylor et al. noted that predicting these dynamic variables is problematic and neglected


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these variables, they did offer a potential solution by using survival analysis to model the

time until a claim is finalised. By relating claim status as being a claim characteristic

(Open or Finalised), other dynamic variables can also be predicted in a similar fashion.

Prevett and Gifford (2007) condensed various dynamic risk characteristics into a value of

single-dimensioned “claim state”, for which a transition model was built.

1.2 Claim Revisions

This paper models the claim revision process. An open claim would have a set of claim

characteristics, against which the claims officer would have put a case estimate. This case

estimate would be what the claims officer thinks the claim could cost. This estimate may

not be a statistical mean of the claims outcome but an estimate based on another criterion.

For example, it could be an estimate at the 75th percentile, so the ultimate claims cost

would be below the case estimate 75% of the time.

Some time later, the claims officer would receive additional information about the claim;

this new set of information would now be the basis of a revised case estimate. The

information may contain updated injury details, the appointment of a plaintiff lawyer or

a court decision regarding the claim. The claims officer now needs to make a few

decisions, including whether or not the claim is to remain open or can now be settled;

whether the new information warrants an upward or downward change and, if so, how

much. Since the case estimate that the claims officer makes would be based on the new

information and prior information, the use of the dynamic claims information would be

insightful to the way claim estimates change.

The complete claim development process is broken down into its component processes.

Four separate processes are specified and modelled:

- Delay processes (denoted by t ) – a continuous or discreet variable that measures the

time until the next change in the case estimate

- Claim status process (denoted by S ) – a binary variable that tracks the claim status of

a claim

- Movement direction process (denoted by D ) – a binary variable that tracks the

direction of the incurred cost change

- Size of change process (denoted by Y ) – a continuous or discreet process that

measures the size of the change.

The modelling of these component processes is conducted in a Generalised Linear

Modelling (GLM )framework. The GLM technique is widely used in the actuarial field

for both pricing and reserving and it is accepted as a robust modelling tool. In addition,

using a GLM framework for these processes enables an easy interpretation of the model

parameters.

It is proposed that the main time measurement will not be in conventional time units, but

rather on a time scale that counts the number of changes since the occurrence of the

claim. Generally, changing claim-handling speed due to staff capacity and claim

frequency will affect the duration of a claim on a calendar time scale. However, if a


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claim’s lifecycle can be thought of as “stages” that prompt new information, the ordinal

number of changes may provide greater correlation with the pattern of claim

development.

1.3 Uses of the Model

By fitting the various models and obtaining model parameters, a simulation model can be

developed to project open claims to their ultimate values. This approach can develop the

case estimate of the open claims to an expected ultimate claims cost and hence can be

used in a pricing exercise along with finalised claims. The simulation model is not dealt

with in this paper. However, by fitting the models and obtaining the parameters,

insights are gained into the “drivers” of claims cost evolution.

Although the initial aim of this project was to be able to model open claims together with

closed claims in a unified framework, the projection of open claims to an expected

ultimate claim size also makes this process a reserving tool. The model, while projecting

claims development through time, would fill out the bottom half of a development

triangle. However, only the Incurred But not Enough Reported (IBNER) component is

projected, with Incurred But not Reported (IBNR) left for some other model.

This approach to valuation is somewhat similar to Statistical Case Estimation (Taylor and

Campbell, 2002; Brookes and Prevett, 2004; and Prevett and Gifford, 2007), where claim

characteristics are used to project claim payments depending on the “state” the claimant

is in for Workers’ Compensation schemes.

With a stochastic model as such, the use of simulation techniques can also produce a

wide range of possible ultimate claims costs, both on an individual claims level and an

aggregated level, potentially adding another measure of volatility to the current staple of

measures.

1.4 Structure

Section 2 describes the data used and a few sample paths of how claims develop. Section

3 takes a brief look at the component processes and proposes the distributions that would

be used to model them. Section 4 specifies the models used and the likelihood of the

claim development process for the chosen distributions and Section 5 presents the results

of fitting the data to the chosen models. Section 6 provides some of the improvements

that can be made and Section 7 provides some concluding remarks. Appendix A

provides the likelihood of the claim development processes and their derivaives;

Appendix B shows which variables are found to be significant in the modelling of the

various processes.


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2 Data

The technique is applied to the NSW CTP industry data. In NSW, CTP is a motor vehicle

insurance product that covers bodily injury arising out of driving, or some other limited

uses, of motor vehicles. The product is considered to be long tail in nature, as the injury

takes some time to stabilise, and the claim settlement process can take time following

injury stabilisation, in particular, if the claim is litigated the court decision process may

take considerable time.

The Motor Accidents Association (MAA) of NSW regulates the CTP scheme in NSW and

keeps a Personal Injury Register to aid pricing, regulation and scheme monitoring. NSW

CTP insurers submit their claims information, payment details and case estimates on a

quarterly basis. The MAA has kindly granted permission for this paper to use the PIR

database to develop the models and to present the results obtained from the data.

The NSW CTP scheme was deregulated with the Motor Accidents Act 1988 and, after

obtaining a licence, private insurers were allowed to participate and issue CTP

“Greenslips”. In 1999, the Motor Accidents Compensation Act (MACA) was introduced.

The analysis in this paper is limited to 7.5 years of data, that is, accidents occurring from

1 January 2000. After the data were “cleaned” (for example Accident Notification Forms

(ANF’s) that did not eventuate into a claim were removed; claims finalised in the quarter

they were reported were also removed) the dataset contains over 71,000 claims, of which

around 60,000 claims are finalised, totalling around 200,000 revisions.

2.1 Sample Paths

The following diagrams present two “sample paths” of claim development. They

provide a simple visualisation of the data.


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Figure 2-1 – Claim Development Sample Path 1

For the claim above, the accident occurred in Q4 2001 but was not reported until Q3 2002.

It then lay dormant for another six quarters; during this period not much was known

about the claim. At the end of 2003, more information about the claim was known

(including that the claimant obtained legal representation and some information

regarding the injury) and immediately the estimate increased to around $10,000. About a

year later, the liability status was changed from “Rejected” to “Accepted” and a further

sizeable increase in the estimated claims cost was made. Over another year elapsed and

the injury seemed to have worsened from one body region to two body regions. The

claim was settled soon after with a saving, almost five years after the claim occurred.

The next claim, however, has had very frequent revisions, almost quarterly. Yet, the main

characteristics of the claim did not change throughout the life of the claim. It was

finalised around six years after it occurred, with three significant upward revisions, but

settled with a saving.


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Figure 2-2 – Claim Development Sample Path 2

3 Exploratory Analysis

In this section, various features of the data, in particular the variable of interest, are

examined and distributions are chosen for the CTP dataset. Other datasets would

probably warrant selecting a different set of assumptions and distributions.

3.1 Claim Development: on Real Time vs. Number of Revisions

One of the features of the model is that the maturity of a claim is not calculated on real

time, but on the number of revisions to claims cost, somewhat analogous to the concept

of “operational time”.

Using the number of revisions as a measurement of time, effects of varying speed of

finalisation, or pace at which the claims are attended to, can be accounted for. For

example, a significant reduction in claims frequency in the NSW CTP scheme occurred

after the introduction of MACA. This may have resulted in, amongst other potential

drivers, claims being attended to more promptly and frequently. From the data, it is

observed that the delays between revisoons are shortening for the more recent years.

Such a proposition would have the effect that claims in more recent accident years are

settled faster; and their development with respect to real time should be larger in the

initial years of development and smaller in the later years relative to the earlier accident

years, as seen from the table below. At the same time, overall average claims sizes are

impacted by settlement of large, if these claims are being brought forward. This could be


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misinterpreted as superimposed inflation (SI), showing excessive levels of SI in an

environment where claims are settled faster.

Table 3-1 – Development Factors by Year

Development Factor (yearly)

1 : 0 2 : 1 3 : 2 4 : 3 5 : 4 6 : 5

2000 119% 113% 120% 104% 98% 101%

2001 121% 120% 113% 101% 100% 97%

2002 128% 116% 115% 101% 99%

2003 126% 120% 114% 95%

2004 130% 116% 112%

2005 141% 114%

2006 138%

Stdev 8.2% 2.9% 3.0% 3.5% 0.8% 2.7%

Acc

Year

*Note, the development factors are the open claims at the end of a year compared to their incurred costs at

the end of the next year. That is, finalised claims are removed when calculating the development factors.

Table 3-2 - Average Size by Settlement Maturity

Development Year

0 1 2 3 4 5 6

2000 1,999 6,084 18,237 44,252 105,733 135,139 199,032

2001 1,424 6,509 25,502 77,779 124,818 165,394 237,442

2002 1,088 8,408 29,088 74,517 134,622 239,123

2003 1,707 9,626 32,886 90,576 163,847

2004 1,850 11,462 45,990 107,282

2005 2,397 13,432 56,913

2006 3,686 16,212

Rate of Inc 12.2% 16.8% 21.7% 19.2% 13.9% 28.5% 17.6%

Acc

Year

The Payments per Claims Finalised in Operational Time model, someimes used for the

valuation of CTP claim liabilities, can be used to account for the increase in the speed of

finalisation. However, the model does not work well when the profile of claims or the

order of claims finalising changes. By modelling claim delays on other claims

characteristics, the changing pattern in finalisation may be better understood and

allowed for.

The development factors based on the number of changes is more stable. In the

following table, the development factors by the number of revisions do not exhibit a clear

pattern like those factors by development year. The variability of the development

factors, as measured by their standard deviation, is comparable between the two time

measures. However, the number of claims used to calculate each cell for the number of

revisions approach is smaller compared to the development by calendar year approach,

especially towards the end for each accident year. Therefore, even though the

comparison understates the stability of the development factors by number of revisions,

the variability of the development factors by number of revisions is small, especially for

the earlier developments.


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Table 3-3 Development Factors by Number of Revisions Development Factor (at ith revision to claims cost)

1 2 3 4 5 6 7 8 9 10

2000 110.1% 108.3% 107.7% 109.8% 110.7% 103.6% 105.0% 104.3% 104.5% 105.0%

2001 111.2% 113.1% 111.8% 105.7% 108.7% 108.2% 100.2% 99.5% 104.1% 100.7%

2002 110.3% 112.5% 109.7% 109.9% 107.6% 105.7% 108.7% 98.9% 102.7% 94.4%

2003 100.6% 117.1% 113.0% 108.5% 107.9% 103.5% 100.6% 99.0% 103.6% 103.3%

2004 109.8% 117.1% 119.7% 115.0% 104.1% 104.4% 98.7% 106.9% 98.7% 84.8%

2005 110.7% 115.7% 114.7% 115.7% 111.5% 107.5%

2006 107.3% 117.6% 118.0% 119.9%

2007 111.5% 109.0%

Stdev 3.6% 3.7% 4.3% 4.9% 2.6% 2.0% 4.1% 3.7% 2.3% 8.2%

Acc

Year

3.2 Delay between Revisions

When using revisions as a measure of time, the actual time between revisions needs to be

analysed. The preferred format of delays between revisions would be the number of

days and the exponential distribution is expected to fit the delay distribution well. The

exponential would also be easy to work with.

The industry data are summarised by quarter and delays are now more discreet in that

they take on values of 1, 2, 3, … (quarters). As such, Geometric and Negative Binomial

distributions are candidates from the main discreet distributions. On the other hand,

some continuous distributions that are expected to fit delays measured in days well are

“discreetised” and compared to the discreet distributions.

Figure 3-1 – Observed vs. Expected Delays – All Data

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Delay (qtrs)

Pro

po

rtio

n

Observed Expon. Gamma Geometric Neg Binom

From the above graph, the Negative Binomial distribution fits best, while the others

performed similarly and underestimated the level of revisions with a delay of one

quarter.


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3.3 Claim Status Process

A variable is used to monitor when a claim would be finalised to end the claim revision

process. Initially, a variable monitoring the number of revisions during the lifetime of a

claim was considered. However, this would be severely limiting as this number of

revisions variable would be determined at the outset of the claim, based on claims

information as at the time of the accident.

An alternative is to use an indicator variable that monitors the current status of the claim.

Such a binary variable would be determined from all the current covariates and would be

more realistic. When modelling a binary variable, it is equivalent to modelling the

probability of finalisation at each revision; and the probability of finalisation can be quite

different at different revisions. However, if the probability of settling is constant, then

the number of revisions until a claim is finalised should follow a Geometric distribution.

The Geometric distribution is found to fit reasonably well, as shown in the following

graph.

Figure 3-2 – Observed vs. Actual Number of Revisions – All Data

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of Revisions

Pro

po

ert

ion

Observed Geometric

Given the observed data would be driven by different means, it would have greater

volatility than the standard Geometric. This is seen in the graph through the higher

proportion of observations with a delay of 1. The complete data has higher variability

than the standard gamma as expected and modelling the number of revisions using a

claim status indicator is appropriate.

3.4 Claim Revision Direction of Change

When the timing of a revision is determined and whether the change is the final

adjustment or is an intermediate revision has been modelled, the final step is to model

the change. It is proposed to model the direction of the change and the magnitude of the

change separately. This is because the change variable displays a highly asymmetrical

distribution (positive change and negative change behave differently) and, therefore,


11

would be difficult to identify an appropriate distribution to model. This is observed in

the industry CTP dataset; however, other datasets may be different, and modelling the

change as a single process may be considered.

The direction of the change will simply be a binary variable which is modelled using

covariates. From the following graph, it can be shown that the final revisions are mostly

negative revisions, while other revisions are mostly positive. The mix of the directions of

changes does not show a trend against the number of revisions.

Figure 3-3 – Proportion of Direction of Change, but Claim Finalisation Status

1 2 3 4 5 6 7 8 9 10

The ith Revision

Neg

ati

ve

Po

sit

ive

Not final, up Final, up Not final, down Final, down

*Bar on the left is revisions prior to finalisation, bar on the right is the revision on which the claim is finalised

3.5 Size of Change

Change is defined, as in the finance literature, as the log of the new incurred cost less the

log of the existing incurred cost. Such definition provides a magnitude of change that is a

strictly positive continuous variable.

The following table shows the distributing of the observed size of changes, rounded

down to the nearest 0.5. It can be seen that while a large proportion of changes follow the

same distribution, negative changes at finalisation seem to behave quite differently.

There is certainly something interesting about these “savings at finalisation” changes.

These savings at finalisation occur over 80% of the time and represent a significant

portion of total changes.


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Figure 3-4 – Observed Changes by Claim Status and Direction of Change

0%

10%

20%

30%

40%

50%

60%

70%

80%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

log(change) - rounded down to next 0.5

Pro

po

rtio

n

Not final, down Not final, up Final, down Final, up

The following graphs show that the Gamma, Weibull and Generalised Gamma

distributions fit the size of change process quite well and almost indistinguishably. For

versatility, it is hence proposed to use the Generalised Gamma distribution, of which the

Gamma and Weibull distributions are special cases, for the extra flexibility when fitting

the data.

Figure 3-5 – Observed vs. Expected Magnitude of Savings on Finalisation

0%

5%

10%

15%

20%

25%

30%

35%

40%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Pro

po

rtio

n

Observed Weibull Lognormal Gen Gamma Gamma


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Figure 3-6 – Observed vs. Expected Magnitude of Other Revisions

0%

10%

20%

30%

40%

50%

60%

70%

80%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Pro

po

rtio

ns

Observed Weibull Lognormal Gen Gamma Gamma

4 Model Specification

4.1 Notation

Let ijX be the claim cost incurred for claim i at its j th revision. The process ijX is a non-

negative process that maps the claims cost over its life. Let 0iX refer to the initial

estimate of a claim when it first comes to the insurer's attention. With the industry CTP

data, this value is not known, and the incurred value at the end of the reported quarter

was used as a proxy.

Let n be the number of claims in the portfolio, hence, , ,...,= 1 2i n . Let ijT be the time of

the j th revision of the i th claim. Also, let iT denote the date of the accident on which

the claim arose. Then, the time when the claim is reported is denoted as 0iT and

= −0 0i i it T T is the reporting delay — the period of time after the accident before the claim

is reported to the insurer. More generally, , --= 1ij ij i jt T T is the amount of time elapsed

since the last revision until the j th revision. Let im be the number of revisions made to

the i th claim before it is finalised, hence, , ,...,= 1 2 ij m .

Also let *iTJ be the largest j for the i th claim such that ijT is less than *T , in other words,

the last revision made to a claim before the time *T . Therefore, *iTiJX is the incurred

claims cost to date for the portfolio at date *T .


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4.2 Conditioning on Past History

The four processes can be interpreted as one event and may be modelled together with a

joint distribution. However, dealing with joint distributions would add significant

complexity to the model; thus the complete change process is broken down to component

processes. Conditioning is introduced between the processes and covariates, effectively

creating a hierarchy of the order in which the processes happen. The conditioning is

explicitly specified in the following fashion.

The hierarchical order of the claim processes is:

1. The delay process; it is assumed to model the time between the current revision and

the previous revision and on a real-time scale it is the delay in the next revision

taking place. This process is conditioned on only the information available at the

previous revision.

2. The claim status process; it is assumed that this process will determine whether a

claim is finalised at the current revision. This process is conditioned on prior

information, current values of covariates and the current value of the delay process.

3. Once the claim status is determined, the next process is the direction of the change.

This process is conditioned on prior information, current values of covariates, as

well as the current values of the delay and claim status processes.

4. Lastly, the claim size movement distribution; the distribution for this process will

be conditioned on past history, current values of covariates, as well as the current

values of the preceding process variables.

For model specifications, the following filters are used. Let ijF be the history of all the

process variables (delay, claim status, direction of change and size of change variables)

up to the j th revision. Let ij

G be the history of all the other covariates up to the j th

revision.

4.3 The Delay Process

The Negative Binomial distribution is adopted for the delay process measured in

quarters. The delay process is conditioned only on prior history, that is, information

known up to the − 1j th revision.

( ) ( )1

, 1 , 1

1 1| , 1

1

−

− −

− + − = = −

−

tr

ij i j i j

t rP t t F G p p

t (1)

For , ,...= 1 2k

However, an alternative parameterisation is adopted to aid the GLM modelling where the mean

parameter µ is specified directly.


15

( )( )

( )

1

, 1 , 11

2| ,

11

µ

µ−

−−

− −− +

− += = − +

ijij

dij

ij

td

d

ij i j i jt e

d

et eP t t F G

te

(2)

where /= 1d r and µ−

=1 p

rp

4.4 The Claim Status Process

Since the claim status can either be “Open” or “Finalised” (complications of reopened

claims are ignored), a Bernoulli process would be the natural selection. Let ij

S be the

claim status process for the i th claim on its j th revision.

1 if the jth revision is the last

0 if the jth revision is not the lastij

S

=

(3)

( ), 11| , ,ij i j ij ij ij

P S F G t q−= = (4)

Based on this formulation, ij

S can be determined based on all the information available at

time , 1i jT − , the new claims information as contained in filter G and the delay until the

current revision, ijt .

4.5 The Direction of Change Process

Similar to the claim status process, the revision direction process is also binary, taking

values up or down. Let ij

D be a Bernoulli variable such that

1 if the th revision is a positive change

0 if the th revision is a negative changeij

jD

j

=

(5)

( ), 11| , , ,ij i j ij ij ij ij

P D F G t S p−= = (6)

4.6 The Size of Change Process

This process models the size of the change as the absolute difference between the logs of

the pre and post revision claim costs; let ij

Y be a non-negative process that measures

( ) ( ), 1log logij i j

X X −− . Changes of less than 1% of the incurred costs are ignored; with

an average claim size of around $100,000, such a threshold would be around $1,000.

A generalised gamma distribution has been chosen for modelling the NSW CTP due to its

versatility of a three-parameter distribution. By programming the likelihood and

maximisation algorithms, the Generalised Gamma can achieve a wide range of

distributions. This can be done relatively easily with the statistical package S-PLUS.

However, due to its data capacity issues, a dataset of around 200,000 observations and 50

covariates cannot be optimised and this approach was therefore abandoned.


16

As such, the standard Gamma distribution has been chosen; from the previous graphs

(Figures 3-4 and 3-5), the performance of the Gamma is virtually identical to the

Generalised Gamma and its density function is defined as:

( )( )

1

, 1

1| , , , ,

ij ij

ij

c y

ij

ij i j ij ij ij ij

ijij ij

yf y F G t S D e

c

λ

λλ

−−

−

= Γ

(7)

Where λij is proportional to the mean of the variable ijy

4.7 The Ultimate Claims Cost Process

Let iX be the ultimate cost of a claim, then using the notations defined above

( )2 1

0

1

ij ij

mD Y

i i

j

X X e−

=

= ∏ (8)

When constructing the likelihood, finalised claims and open claims need to be treated

separately. This is because the entire history of movements of closed claims is known;

however, for open claims, only the movements up to *iTJ are know, plus the fact that

there have been no movements between *T and *iTiJT .

Hence, the likelihood for the i th claim can be expressed as

( ) ( ) ( ) ( ) ( ) ( ) ( )( )* *, ,, , , , , *λ µ

<= ⋅ < −∏ iT iTi

ij ij ij ij i J i Jj mj

L p q c d P t P S P D f y I P t T T (9)

Where ( )condI is an indicator function that takes on the value 1 if (cond) is true and 0

otherwise.

The likelihood can be expressed in this way due to the conditioning that is assumed in

each of the processes as described above; notations for this conditioning are dropped

from the likelihood to keep the equation manageable.

4.8 Variables and Regression

Three types of variables are used to model the above processes. These are slightly

different from the three categories defined by Taylor et al, 2006.

1. Process variables themselves, that is, the current values of the four processes, are

modelled upon past values of themselves as well as other processes where

appropriate.

2. Static variables, or baseline variables, do not change throughout the duration of the

claim. These variables may include age of claimant at date of accident, gender,

location of the claimant, etc.


17

3. Dynamic variables, change during the evolution of the claim. The changes in these

dynamic variables are the most interesting in determining the outcome of the

process variables. The dynamic variables include Liability Status, Injuries and their

Severities, Legal Representation, etc.

Let X be a vector of covariates for claim i ; for ease of notation, the subscript i is left out.

Then, the following relationships are assumed relating the covariates to the mean

functions of the processes. These follow the standard relationships in a GLM framework.

( )

0

0

0

0

log

log1

log1

log

T

ij T

q q q q

ij

ij T

p p p p

ij

T

ij

X Z

qX Z

q

pX Z

p

X Z

µ µ µ µ

λ λ λ λ

µ α α

α α

α α

λ α α

= + =

= + = −

= + = −

= + =

(10)

4.9 Modelling Software

SAS and S-PLUS are two of the more well-known statistical and data modelling software

programs. SAS (version 9.1 and earlier) has an extensively range of modelling methods

that, while flexible, are still limited to the options available. S-PLUS, on the other hand,

offers a programming environment with which users can code tools that are tailored to

the user’s needs.

The above distribution functions and likelihoods in the previous sections are developed

so they can be programmed into S-PLUS to optimise the likelihood directly to estimate

parameters. However, S-PLUS has been found not to be able to handle datasets of even

moderate size. The models were fitted with the standard SAS GENMOD Procedure and

its various limitations are inherited.

However, SAS 9.2 should bring the welcomed addition of user programming

functionality. This enables the fitting of these models through likelihood maximisation;

as such, the various likelihood functions and their derivatives can be found in Appendix

A. Undoubtedly, there is other software beside S-PLUS and SAS 9.2 that would be able

to carry out likelihood maximisation.


18

5 Results

Given its ability to handle large datasets, the models are fitted using SAS. For the open

claims, an adjustment has to be made in accordance with Formula (9), for the fact that

they have not had a revision in the period of time since the last revision. This is done

through “weighting” the observation when modelling the delay process.

The full set of variables used, with only the main effects fitted, is contained in Appendix

B. Those that showed a high level of significance (with a p-value of <0.01) are marked

with an “*”. The fitting used a step-wise variable selection process where the full set of

variables is used and insignificant ones are gradually dropped out.

The coefficients of a few of the variables that are of interest are presented below. In the

tables, “Type” denotes the type of variable:

� D – Dynamic variable

� S – Static variable

� P – Process variable

The coefficients are presented for:

� t_ij – the delay process

� S_ij – the finalisation process

� D_ij – the direction of change process

� Y_ij – the size of change process, modelled separately for positive and negative

changes

The delay process is modelled on information available at the previous revision only.

Hence, its coefficents of the dyamic variables are based on the values at the previous

revisions. The coefficents of the dynamic variables for the other processes are based on

their current values.


19

5.1 Ith Revision vs. Development Year

Table 5-1 – Model Output – Ith Revision Variable

D_ij = 0 D_ij = 1

D Ith Revision 1 0.50 0.25 1.26- 0.32 0.55-

D Ith Revision 2 - - - - -

D Ith Revision 3 0.11- 0.01 0.25 0.10- 0.00

D Ith Revision 4 0.17- 0.08 0.36 0.16- 0.00-

D Ith Revision 5 0.23- 0.17 0.44 0.23- 0.01

D Ith Revision 6 0.25- 0.21 0.49 0.23- 0.04-

D Ith Revision 7 0.28- 0.36 0.45 0.27- 0.07-

D Ith Revision 8 0.29- 0.33 0.34 0.23- 0.09-

D Ith Revision 9 0.26- 0.41 0.51 0.21- 0.14-

D Ith Revision 10 0.29- 0.41 0.30 0.32- 0.16-

D Ith Revision 11 or more 0.28- 0.43 0.73 0.35- 0.22-

Y_ij Type Parameter Level1 t_ij S_ij D_ij

*The second revision is taken as the baseline because the first revision does not use the full set of variables, as

it lacks variables at the “previous revision”. Because of this, the coefficients for the first revision are on a

different level.

The modelling shows that the delay between revisions shortens and the probability of

finalising the claim increases with the increasing number of revisions. The likelihood of

having a positive change also increases; however, the change is more likely to be of

smaller size, for both positive and negative changes. Putting it all together:

� Earlier revisions tend to be quite spaced out, with bigger movements

� Later revisions (as measured by the number of revisions) tend to be close together;

and are more likely to be small upward adjustments

The variable “development year”, which is defined as (year of revision – year of

accident), was added to test whether calendar year development can add extra

explanatory power to each of the models. It is of interest to note that the number of

revisions variable has considerably more explanatory power compared to development

year in all processes except for the finalisation process. The time elapsed since the date of

accident is still a significant driver for the probability of finalising the claim at a

particular revision.


20

5.2 Injury Severity

Table 5-2 – Model Output – Injury Variables

D_ij = 0 D_ij = 1

D ISS unknown/admin . - - - .

D ISS 1 . 0.30 0.40- . .

D ISS 2 . 0.21 0.34- . .

D ISS 3 - 5 . 0.09 0.19- . .

D ISS 6 - 10 . 0.03- 0.04- . .

D ISS 11 - 30 . 0.05- 0.01 . .

D ISS 31 - 75 . 0.42 0.12 . .

D ISS increase 1 . 0.38 . . .

D ISS increase 0 . - . . .

D Number of Regions 0 . - - -

D Number of Regions 1 . 0.12 0.05- 0.10





D Num Reg increase 1 . 0.48 0.18- 0.10

D Num Reg increase 0 . - - -

Type Parameter Level1 t_ij S_ij D_ij Y_ij

It appears that light injuries and heavy injuries (as measured by the Injury Severity Score

(ISS)) are the most likely to finalise at a particular revision; this is especially so if the ISS

was increased at the revision. This may be counter intuitive, as it is generally accepted

that claims from severe injuries takes longer to settle. However, possible explanations for

this include a small number of observations in the group with an ISS of 31 to 75.

Secondly, there is a general tendency for ISS to worsen during the life of a claim and the

higher ISS may represent records from the later revisions of a claim and, hence, there

may be an interaction effect not accounted for in the modelling.

Similarly, a higher number of body regions injured also shows higher likelihood of

finalisation, as well as for an increase in the number of regions injured. Again, possible

explanations may include:

� A small number of records with 4 or 5 regions injured and also a small number of

records that had an increase in the number of body regions at a particular revision,

and coefficient may be volatile

� More severe injuries may be correlated with larger claims and the fitted models

may not account for the interaction effect

More severe injuries are also more likely to have an upward revision compared to less

severe ones. On the other hand, more regions of injury tend to have a relatively lower

chance of an upward revision; however, when they do, the magnitude of these revisions

tends to be bigger.


21

5.3 Delay between Revisions

Table 5-3 – – Model Output – Delay Between Revision Variable

D_ij = 0 D_ij = 1

P t_0 1 . - - - -

P t_0 2 . 0.18- 0.11 0.20 0.04-

P t_0 3 . 0.06- 0.17 0.35 0.04

P t_0 4 . 0.06- 0.25 0.42 0.09

P t_0 5 . 0.03- 0.25 0.48 0.13

P t_0 6 . 0.05 0.35 0.51 0.17

P t_0 7 . 0.01- 0.38 0.53 0.19

P t_0 8 . 0.09 0.35 0.56 0.17

P t_0 9 . 0.04 0.49 0.52 0.22

P t_0 10 . 0.12 0.47 0.56 0.21

P t_0 11 - 15 . 0.11 0.60 0.54 0.23

P t_0 16 or more . 0.01- 0.74 0.49 0.29

P t_1 1 - - - - -

P t_1 2 0.07 0.10 0.01- 0.10 0.07

P t_1 3 0.11 0.27 0.00 0.14 0.11

P t_1 4 0.13 0.30 0.04 0.15 0.15

P t_1 5 0.14 0.31 0.17 0.10 0.19

P t_1 6 0.11 0.38 0.14 0.12 0.23

P t_1 7 0.10 0.39 0.21 0.10 0.24

P t_1 8 0.06 0.45 0.32 0.17 0.25

P t_1 9 0.02 0.45 0.30 0.09 0.30

P t_1 10 0.01- 0.52 0.22 0.02 0.29

P t_1 11 - 15 0.01 0.64 0.27 0.06 0.27

P t_1 16 or more 0.06 0.46 0.25 0.00- 0.14

Type Level1 t_ij S_ij D_ij Y_ij

Parameter

* Coefficients for t_ij only exist for t_1 as the length of the current delay is modelled on the length of the

previous delay

The delay process variables ijt and , 1i jt − are extremely useful in the modelling of the

process of interest. Typically, longer delayed revisions would lead to:

� being more likely to finalise,

� being more likely to be an upward revision, and

� a larger size of change, for both upward and downward changes.

These observations also apply for the delay from the previous revision.


22

5.4 Year Effects

Two year effects are presented below; year of accident and year of revision. The

coefficient of the delay process shows that claims from the more recent accident years are

subject to shorter delays and similarly for the revisions made in the more recent years.

This pattern may be caused by the fact claims with revisions from the recent years tend to

be smaller claims. This pattern remains even after the impact of claim size is accounted

for.

However, for probability of finalisation, the coefficients between accident year and

revision year are negatively correlated, suggesting a “development year effect”. In fact,

when a development year variable was added to the model, its explanatory power

outweighed that of the number of revisions variable. For the ijD process, the claims from

more recent accident years are more likely to increase than decrease.

Table 5-4 – Model Output – Year Variables

D_ij = 0 D_ij = 1

S Year of Accident 2000 0.27 1.72 0.16- . .







S Year of Accident 2007 - - - . .

D Year of Revision 2000 0.59 2.40- . . 0.09-





D Year of Revision 2005 0.32 0.81- . . 0.02

D Year of Revision 2006 0.24 0.41- . . 0.03

D Year of Revision 2007 - - . . -

Type Parameter Level1 t_ij S_ij D_ij Y_ij


23

6 Further Improvements

6.1 Limitations of Using Industry Data

For this paper, the claim development processes are applied to the NSW CTP industry

data. There are many reasons why this model would be more informative and powerful

if it were applied to a particular insurer’s dataset.

Firstly, the model accounts for how claim estimates change and this is related to the

claim-handling procedures of each of the insurers. Using industry data, however, the

various claim-handling procedures are inseparable in the data. Thus the modelled effects

will combine all the different insurers’ practices and their explanatory power will be

reduced. A single insurer’s data would be more homogenous and when interactions with

the year variable are specified, it can reveal the changes in claim-handling procedures.

Secondly, industry data are summarised quarterly. Since it is not known whether there

was a revision in the same quarter that a claim was reported, the initial information

regarding the claim is not known; such information may be useful and add further

explanatory power. Also, with the data summarised quarterly, the delay variable was

modelled using a discreet distribution. Discreet distributions are generally more difficult

to work with when accounting for the adjustment for the open claims.

Thirdly, the individual insurers’ database may contain claim-management indicators.

These variables may include the insurer’s legal representation or claim-handling

approach. The inclusion of these variables may model the effectiveness (and impact) of

variables on claim-management approaches and allow decisions to be made regarding

claim handling for the benefit of both claimants and insurers.

6.2 Modelling Enhancement

Due to the paper’s main aim of presenting a method of modelling the claim development

process by breaking it down, time has not been invested in coming up with perfect

models. Generally, while interaction effects can offer great insight into how claims

develop and how various types of claims are handled, this avenue has not been pursued

for this paper. However, based on the industry dataset, with the possibility of extending

the dataset using claims occurred before 2000, a vast amount of useful information can be

gained.

For projection purposes, dynamic variables are difficult to use. To project or simulate

process variables, the values of the dynamic variables will need to be projected first.

Typically transition models need to be built for each dynamic variable, which is not an

easy task if each one is allowed to be time-dependent or state-dependent.

One possible solution is to turn the dynamic variables into static variables by sampling

their value at a common point, for example, one year after the date of the accident.

Hopefully, by that stage the injuries would have stabilised and legal issues may have


24

progressed such that movements in these dynamic variables will be minimised. One

trade-off to consider when selecting what timeframe to use is that the longer the

timeframe, the better the prediction will be; yet, at the same time, more claims would not

be able to be modelled because these claims are too recent.

7 Conclusions

By breaking up the claim evolution process into its component processes, this paper

presents a novel way of modelling claim developments. This approach, when combined

with simulation, was considered to project open claims for pricing purposes. However,

in doing so, outstanding claims liabilities can be estimated. It is useful to be able to

project claims outstanding on a claim-by-claim basis, as profitability by segment can then

be analysed; this has always been difficult with a long-tailed insurance product.

Actuaries generally prefer not to apply statistical black boxes. Some of the stated dangers

of doing so include that these models cannot account for changing trends effectively and

transparently. With this model, the effects of the variables need to be tracked through

time carefully in order to understand the extent of their impact through time. The benefit

of this is that the drivers of change through time are better understood.


25

8 Bibliography

Brookes, R. and M. Prevett, 2004, Statistical Case Estimation Modelling - An Overview of the

NSW WorkCover Model, Presented to the IAA Accident Compensation Seminar,

2004

McGuire, G., 2007, Individual Claim Modelling of CTP Data, Presented to the IAAust XIth

Accident Compensation Seminar, 2007, Melbourne

Prevett, M. and D. Gifford, 2007, Statistical Case Estimation for Long Term Claimants –

Uncovering Drivers of Long Term Claims Cost in Accident Compensation, Presented to

the IAAust XIth Accident Compensation Seminar, 2007, Melbourne

Taylor, G. and M. Campbell, 2002, Statistical Case Estimation, Research Paper Series,

Centre of Actuarial Studies, The University of Melbourne, No 104

Taylor, G. and G. McGuire, 2004, Loss Reserving with GLM’s: a Case Study, Presented to the

Spring 2004 Meeting of the Casualty Actuarial Society, Colorado Springs,

Colorado

Taylor, G., G. McGuire and J. Sullivan, 2006, Individual Claim Loss Reserving Conditioned by

Case Estimates, Research Paper Series, Centre of Actuarial Studies, The University

of Melbourne, No 151

Part III Appendices

A Likelihoods, First Order Derivatives and Second Order Derivatives

A.1 Likelihoods and log-likelihoods of open and closed claims

The likelihood and log-likelihoods to the closed claims are

( ) ( ) ( )( )

( )( ) ( )

( )

( )

1

1 1

1

1

11

1, , , , , 1 1

2 !

1 ! 1 ! 1

λλ µ

λλ

µ

µ−

−−− −

=

−−

− − +=

= − − Γ

− +⋅

− − +

∏

∏

ij ij

ij ijij ij ij

ijij ij

dijij ij

ij

c ym

S DS D ij

C ij ij ij ij

j ijij ij

td d

mij

d t edj ij

yL p q c d q q p p e

c

t e e

t e e

(1)

and

( ) ( ) ( ) ( ) ( )

( )( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

1

1

2

1 0

1

, , , , , 1 log 1 log log 1 log 1

log log 1 log log

log log 1 !

1 log 1 log 1

λ µ

λ λλ

µ µ

=

=

−−

= =

−

=

= − − + + + − −

+ − Γ − + − − −

+ + − −

+ − − − + +

∑

∑

∑ ∑

∑

�

ij

ij ij ij

m

C ij ij ij ij ij ij ij ij

j

mij

ij ij ij ij ij

j ij

m td

j h

md d d

ij ij ij ij

j

p q c d S q S q D p D p

yc c y

e h t

t e t e e

(2)

The likelihood and log-likelihood of the open claims are

( ) ( ) ( )( )

( )( ) ( )

( )

( )( )( )

*

*

* *

1

1 1

1

1

, 11

1

1, , , , , 1 1

2 !1 *

1 ! 1 ! 1

λλ µ

λλ

µ

µ−

−−− −

=

−−

+− − +=

= − − Γ

− +⋅ − ≤ −

− − +

∏

∏

ij ijiT

ij ijij ij ij

ijij ij

iT

dij iT iTij ijij

c yJ

S DS D ij

O ij ij ij ij

j ijij ij

td d

Jij

t i J iJd t edj ij

yL p q c d q q p p e

c

t e eP t T T

t e e

(3)

where ( )( )

( ) ( )( )

( )* *

*

1

, 11

*

2 !1 *

1 ! 1 ! 1

µ

µ−

−−

∞

+ − − += −

− +− ≤ − =

− − +∑

ijij ij

dijiT iT ij ijijiJiT

td d

ij

t i J iJ d t edk T T ij

t e eP t T T

t e e


and

( ) ( ) ( ) ( ) ( )

( )( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

*

*

*

1

1

2

1 0

, , , , , 1 log 1 log log 1 log 1

log log 1 log log

log log 1 !

1 log 1 log 1

λ µ

λ λλ

µ µ

=

=

−−

= =

−

= − − + + + − −

+ − Γ − + − − −

+ + − −

+ − − − + +

∑

∑

∑ ∑

�

iT

iT

iT

ij

ij ij ij

J

O ij ij ij ij ij ij ij ij

j

Jij

ij ij ij ij ij

j ij

J td

j h

d d d

ij ij ij ij

j

p q c d S q S q D p D p

yc c y

e h t

t e t e e

( )( ) ( )

( )

( )

*

*

1

1

1*

2 !log

1 ! 1 ! 1

µ

µ−

=

−−

∞

− − += −

− +

+ − − +

∑

∑

iT

ijij ij

dijij ij

ijiJiT

J

td d

ij ij

d t edk T T ij

ij

t e e

t e e

(4)

A.2 Likelihood of the claim development process expressed in linear predictor

( )

( ) ( )

( )( ) ( )( )

( ) ( )( )

( ) ( ) ( ) ( )

( )

( )*

2,

0

log 1 log 1

log 1 log log

, , , , ,

log log 1 !

1 log 1 log 1

2 !log

λ

µ µ

λ λ

λ µ−

−

=

−

−

<

− + + − + + Γ − + − − − =

+ + − −

+ − − − + +

− ++

∑∑

�

q p

ij

ij ij ij

ij

iT

Z Z

ij q ij p

ij

ij ij ij Z

tdi j

h

d Z d d Z

ij ij

d

ij

J m

ij

S Z e D Z e

yc Z c y Z

ep q c d

e h t

t e t e e

t eI

t( ) ( )( )

( )*

1

1* 1 ! 1 ! 1

µ

µ

−

−

∞

− − += −

− − +

∑ij

ij

dijij ij

ijiJiT

td Z

d t ed Zk T T

e

e e

(5)

where ( )*iTJ mI

< is an indicator function that takes the value of 1 if the claim is open and 0 if

claim is closed at the end of the analysis period.

With the delay process modelled with a negative binomial distribution, the likelihood

would be too difficult to differentiate with respect to ijd and µα , and then find the

solutions through the Newton-Raphson method. The last part of Equation (5) is known

as the Incomplete Beta function and its derivatives would need to be calculated by

numerical methods. It is processed using an iterative approximation process where the

last part of Equation (5) is not fit using maximum likelihood. The process would be

� Fit the parameters ignoring the Incomplete Beta function

� Using these fitted parameters and estimate the value of the incomplete beta

function


� These evaluated incomplete beta functions is then used to weight the affected

observations at the next fitting of the parameters, and so on.

On the other hand, if the exponential distribution was assumed for the delay process, the

likelihood would be much simpler to fit using maximum likelihood methods

A.3 Derivatives of the log-likelihood

First order derivatives

( ) ( )

( ) ( ) ( )

( )

,

2

, 0

,

1

1 11

1 1 log 11

1

1

1 1

µ

µ

µ

µ

µ

µ

µ

µ

α

α

−

−−− −

−=

−

∂= − − − + ∂ +

∂= − + − − − + − + ∂ + +

∂= −

∂ +

= − − +

∑

∑ ∑

∑

�

�

�

ij

ij

ij

ij ij

ij ij ij

ij ij

q

q

q

d Zd

ij ij ij d Zi j

d d Ztd d d Z

ij ijd d Zi j h

Z

q ij qZi jq

Z

ij q

i

et d t e X

e

e et Z t e e e

d e h e

e X S Xe

S e X

( )

( )

( ) ( )

,

1

,

,

,

1 1

log

λ

λ

λ

λ

α

α

−

−

∂ = − − + ∂

∂= − +

∂

∂= − + −

∂

∑

∑

∑

∑

�

�

�

p

j

Z

ij p

i jp

Z

ij

i j

ij

i j

D e X

c y e X

digamma c y Zc


Second order derivatives are

( ) ( )

( ) ( ) ( )

( )( )

2 22

2,

2 2

,

2 2 2

20

1 1

1 1 11

1 1 1 1 1

µ µ

µ

µ µ

µ

µ µ

µ

µ

µ

µ

α

α

−−

−− −

− −−−

−=

∂= − + +

∂

∂ = − − − + + + ∂ ∂ +

∂ = − − − + + − − +

∂ +

∑

∑

∑

�

�

�

ij ij ij

ij

ij ij ij ij

ij

ij ij ij

ij

d d Z d Z

ij

i j

d Zd d Z d Z d

ij ij d Zi j

td Z d Z d dd

ijdh

t e e e X

et t e e e e X

d e

ht e e e e e

d e h( )

( )

( )

( )

( )

( )

1

,

,

2 1

2,

22

,

2 22

2,

22

2,

2

2,

2

,

log 11

1

1

1

µ

µ

µ

µ

λ

λ

λ

λ

λ

α α

α

α

α

−

− −

−

−

−

−

+ + + +

∂ ∂= − +

∂ ∂

= +

∂= +

∂

∂= −

∂

∂= −

∂

∂= −

∂ ∂

∑

∑

∑

∑

∑

∑

∑

∑

�

�

�

�

�

ij

ij

ij ij ij

ij

q

q q

p p

Z

i j

d Zd d d Z

d Zi j

Z

q

i jq q

Z Z

q

i j

Z Z

p

i jp

Z

ij

i j

i j

i j

ee e e

e

e X

e e X

e e X

y e X

trigamma cc

Xc

The other second order derivatives are zero.

A.4 Using the Generalised Gamma Distribution for Size of Change

If the generalised gamma distribution was used rather than the standard gamma, then

the relevant likelihoods and derivatives are as follow. An exponential version of the

parameters are used for fitting due to the strictly positive requirements of the c and k

parameters.

( )( )

1

, 1| , , , ,λ

λλ

−−

−

= Γ

kijijij ij

ij

yc k

ij ij

ij i j ij ij ij ij

ijij ij

k yf y F G t S D e

c

Let

k e

c e

κ

γ

=

=

( ) 0log λ λ λ λλ α α= + =T

ij X Z


Then the log-likelihood is

( ) ( )( ) ( ),

, , log log log

e

ij

ij ij Zi j

ye e y Z y

e

κ

λ

γ γ κλκ γ λ κ +

= − Γ + − − −

∑�

The relevant first order derivatives are

( )

( ) ( )( )

( ) ( )

,

,

,

1 log log

log

e Ze

ij

i j

eZ Z

ij ij ij

i j

ij

i j

e e y e X

e y Z y e y e e

digamma e e e y Z

κκλ

κ

λ λ

γ κ κλ

λ

γ κ κλ

γ γ γ κλ

α

κ

γ

−+

+

+

∂= − +

∂

∂= + − −

∂

∂= − + −

∂

∑

∑

∑

�

�

�

The second order derivatives are

( ) ( )( ) ( )( )

( ) ( )( ) ( )

( ) ( ) ( )

22 2

2,

2

2,

2

2,

22

,

2

log log 1 log

log

log

e Ze

ij

i j

eZ Z Z

ij ij ij ij

i j

ij

i j

eZ Z Z

ij ij ij

i j

e y e X

e y Z y e e y e y e e

e trigamma e e digamma e e y Z

e e y e e y e y e e X

κκλ

κ

λ λ λ

κ

λ λ λ

κλ

λ

γ κ κ κλ

γ γ γ γ γ κλ

γ κ κ κ κλ

λ

α

κ

γ

α κ

α

−

+

+

+

∂= −

∂

∂= − − +

∂

∂= − + + −

∂

∂ = − + +

∂ ∂

∂

∂

∑

∑

∑

∑

�

�

�

�

�

( )

,

2

,

log

i j

ij

i j

e X

e y Z

γ κλ

λ

γ κλ

γ

γ κ

+

+

= −∂

∂= −

∂ ∂

∑

∑�


B Model Outputs – Significant Variables

D_ij = 0 D_ij = 1

Dynamic Brain * *

Eco Loss *

ISS * *

ISS increase *

Ith Revision * * * * *

Leg Rep increase * * *

Leg Representation * * * *

Liability * * * * *

Liability increase * *

Lit Level increase * * *

Litigation *

Litigation Level * * *

Max Sev increase *

Max Severity *

Number of regions * * *

Num Reg increase * * *

RegionG Inj * * *

Rehab * * *

Rehab increase

Spine *

Whiplash *

Year of Revision * * *

Process D_0 *

D_1 * *

S_0 *

t_0 * * * *

t_1 * * * * *

X_1 * * * *

Y_1 * * * *

Static Age at Acc. * * * * *

Employment Status * *

Gender * * *

Year of Accident * * *

Region of Risk

Vehicle Category

Type Variable t_ij S_ij D_ij Y_ij

Modelling Claim Development Processes - Actuaries Institute Wang_Claim... · 2010-05-18 ·...

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